Verified solution · Exam P

SOA Exam P Sample Question #110 Solution

This Exam P sample reference tests Joint Probability Decomposition. Exactly two losses can occur through year-count pairs (0,2), (1,1), or (2,0). Their joint probabilities sum to 0.13125, choice E.

Independent analysis by ActuaryProof · reviewed August 4, 2026

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Independent solution

How to solve this Joint Probability Decomposition question

Setup

Exactly two losses can arise only from the three ordered year-count pairs listed in the next step. These cases are disjoint and therefore their probabilities may be added.

\Pr(X=x)=0.5^{x+1},\quad X+Y=2

Model

For each admissible first-year count, multiply its marginal probability by the corresponding conditional probability for the second-year count.

(X,Y)\in\{(0,2),(1,1),(2,0)\}

Compute

The three joint contributions are 0.025, 0.075, and 0.03125. Their sum is 0.13125.

\Pr(X+Y=2)=0.5(0.05)+0.25(0.30)+0.125(0.25)=0.13125

Answer

The two-year total equals two with probability about 0.131.

\boxed{0.131\quad\text{(E)}}

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Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

AThe value 0.025 includes only the ordered pair with zero losses in the first year and two in the second.
BThe value 0.031 includes only the ordered pair with two losses in the first year and none in the second.
CThe value 0.075 includes only the ordered pair with one loss in each year.

Original practice · fully worked

Original variant: one incident across two reporting periods

The first-period incident count has probability P(X=x)=0.6(0.4)ˣ for nonnegative integers x. Also, P(Y=1|X=0)=0.20 and P(Y=0|X=1)=0.50. Find the probability of exactly one incident across both periods.

A
0.12
B
0.18
C
0.24
D
0.30
E
0.36

Variant answer in brief

Only pairs (0,1) and (1,0) contribute. Their probabilities are 0.6(0.2)=0.12 and 0.24(0.5)=0.12, totaling 0.24, choice C.

Setup

A total of one incident can occur only through the ordered pairs zero then one, or one then zero.

\Pr(X=0)=0.6,\quad \Pr(X=1)=0.24

Model

Compute each joint probability as a first-period probability multiplied by the appropriate conditional second-period probability.

\Pr(X+Y=1)=\Pr(0,1)+\Pr(1,0)

Compute

Each of the two disjoint paths has probability 0.12, so their sum is 0.24.

\Pr(X+Y=1)=0.6(0.20)+0.24(0.50)=0.24

Answer

The total-one probability is 0.24.

\boxed{0.24\quad\text{(C)}}

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Cite this solution

ActuaryProof. “SOA Exam P Sample Question #110 — Verified Solution.” Verified 2026-07-31.

Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.

How to solve this Joint Probability Decomposition question

Setup

Model

Compute

Answer

All 718 Exam P sample solutions in syllabus order

Why the other letters tempt

Original variant: one incident across two reporting periods

Variant answer in brief

Setup

Model

Compute

Answer

Related Exam P solutions

Turn this solution into a complete study sequence

Cite this solution